What is the smallest number by which 3600 be divided to make it a perfect cube?a)9b)50c)300d)450Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Prime factorise 3600.

You will get 3600 = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5

Now group the numbers into groups of 3.

You will get 1 group of 2 and remaining numbers i.e. 2,3,3,5,5

Now multiply these extra numbers

You will get 2 x 3 x 3 x 5 x 5 = 450

So, 3600 divided by 450 = perfect cube (8)

Answer

3600=24×32×523600=24×32×52

Therefore, the smallest number by which 3600 needs to be divided to make it a perfect cube
=2×32×52=2×9×25=450

Answer

Given:
3600 is the number that needs to be divided to make it a perfect cube.

To Find:
The smallest number by which 3600 can be divided to make it a perfect cube.

Solution:
To find the smallest number by which 3600 can be divided to make it a perfect cube, we need to consider the prime factors of 3600.

Prime Factorization of 3600:
3600 = 2^4 * 3^2 * 5^2

Perfect Cube:
A perfect cube is a number that can be expressed as the cube of an integer.

To make 3600 a perfect cube, we need to divide it by a number such that the powers of all its prime factors are multiples of 3.

Minimum Powers:
To find the minimum powers of the prime factors of 3600 that are multiples of 3, we need to divide the powers by 3 and take the ceiling value.

2^4 / 3 = 1.33 (ceiling value = 2)
3^2 / 3 = 1
5^2 / 3 = 1.33 (ceiling value = 2)

Modified Prime Factorization:
3600 = 2^2 * 3 * 5^2

Now, we need to multiply the modified prime factors to get the smallest number by which 3600 can be divided to make it a perfect cube.

Smallest Number:
Smallest number = 2^2 * 3 * 5^2
= 4 * 3 * 25
= 300

Therefore, the smallest number by which 3600 can be divided to make it a perfect cube is 300. Hence, the correct answer is option (c) 300.

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